We decompose the primitive lattice point counting function into an interior sum and a boundary sum using the coordinates s = m+ n and t = (m−n) / (m+ n). The boundary error ER (x) = sumₛ e (s, x) splits naturally by the primality of s: prime values contribute mostly positive errors, composite values mostly negative ones. Their near cancellation veried exactly to x = 10¹4 with cancellation up to 99. 9% governs the exponent in the Gauss circle problem. This decomposition makes the cancellation mechanism arithmetically transparent, in contrast to the classical Voronoi and χ4-approaches where the same cancellation is analytically present but arithmetically opaque.
Arno Wilhelmsen (Sat,) studied this question.